*3.1. Flow Characteristics*

Figure 5 shows the velocity streamlines and total pressure contours at different locations in the span-wise direction of the R1 blade, to illustrate the effects of different blade damage locations on the flow field. It can be seen that the damage mainly affected the flow field on the suction side of the blade. The hub +5% location had a small flow-circulation zone in the reference case. However, in the damaged blades, this zone was not present, due to the effects of the altered the blade profile on the flow in span-wise direction. In the mid-span, in comparison with the reference case, the middle-damage cases exhibited more circulation, while the top-damage cases exhibited less circulation. As a result, a noticeable difference in the total pressure contours at the mid-span can be seen among the five cases, as shown in Figure 5b. The middle-damage cases exhibited lower total pressures than other cases at the mid-span since they had more flow circulation at this location. Moreover, at the shroud −5% location, a lower total pressure can be observed in the top-damage cases than in the other cases; this is because a small flow circulation was generated in the top-damage cases. It can be concluded that the damage location affected not only the circulation zones at the suction side of the blade but also the total pressure near the damage, which significantly impacted the flow and heat transfer characteristics.

**Figure 5.** (**a**) Velocity streamlines; (**b**) total pressure contours at different span-wise locations of the R1 blade.

Figure 6 shows the temperature contours on the rotor blade surface under various blade conditions. The highest temperature on the pressure side of the blade was near the mid-span of the leading edge, which exhibited more vorticity and was affected by the HS condition. The damage did not affect the temperature distribution on the pressure side, but it significantly affected the temperature distribution on the suction side. The high-temperature regions extended in the span-wise direction in the top-damage cases, and in both the span-wise and radial directions in the middle-damage cases. As shown in Figure 6, the suction-side surface of the damaged blades had a higher temperature than the corresponding surface of the reference blades, due to the effects of the damage on the passage flow and flow circulation. As a result, both the average and maximum temperatures of the blade surface increased when the blades were damaged, as shown in Figure 7. Compared with the reference case, the middle-damage case at the suction side exhibited a higher temperature—by approximately 5 (average) and 2 K (maximum). Considering the effects of the HS, the maximum temperature in the middle-damage cases was higher than that in the other cases.

**Figure 6.** Temperature distribution on the R1 blade under various blade conditions.

**Figure 7.** Average and maximum temperatures on the R1 blade under various blade conditions.

It is necessary to examine the effects of damage locations on the flow characteristics downstream of the rotor blade. Figure 8 shows the contours of static entropy and total pressure at the R1 outlet under various blade conditions. The static entropy was directly affected by the rotor blade conditions. Compared with the reference case, in the top-damage cases, the high-static-entropy regions extended in the radial direction, while in the middle-damage cases, these regions extended in both the span-wise and radial directions. These conditions strongly influenced the temperature distribution, which significantly affected the flow and heat transfer characteristics of the S2 vane. Similarly, the total pressure at the R1 outlet was significantly dependent on the blade conditions. Overall, the total pressure increased when damage occurred on the blade. The increase in total pressure resulted in an increase in the leakage flow passing through the blade tip or the main passage flow. This increment is reflected in the contours of the total pressure, shown in Figure 8b.

The attributes of heat transfer are strongly affected by the flow vortex structure [30,31]. Touil and Ghenaiet [32] investigated the effects of blade–vane interaction on the vortex structure in high-pressure gas turbines. Wei et al. [33] describe the flow structure using an iso-surface with the λ2—criteria method. Figure 9 shows the 3-D complex vortex structure of the flow passing through R1 under various blade conditions. The structure was expressed using the λ2—criteria method, with the magnitudes of strength level and values of λ2 being 10<sup>4</sup> and 5.14 × 10<sup>6</sup> s<sup>−</sup>2, respectively. In comparison with the reference case, the top-damage cases exhibited a weaker tip leakage vortex, while the middle-damage cases exhibited a stronger tip leakage vortex. The pressure field directly affected the tip leakage flow conditions since the tip leakage flow is driven by the pressure difference between the pressure and suction sides of the rotor blade.

**Figure 8.** (**a**) Static entropy; (**b**) Total pressure at the R1 outlet under various blade conditions.

**Figure 9.** 3-D vortex structure at outlet of the R1 blade.

Figure 10 shows the pressure difference between the pressure and the suction sides of the R1 blade and the leakage flow passing through blade tip under various blade conditions. The top-damage cases had lower pressure differences, while the middle-damage cases had higher pressure differences than the reference case. As a result, the tip leakage flow in the middle-damage cases was higher than that in the reference case, whereas the opposite was true for the top-damage cases. The tip leakage flow significantly affected the heat transfer characteristics and efficiency of the gas turbine, as discussed in the following section.

**Figure 10.** Pressure difference between pressure side and suction side of the R1 blade and tip leakage flow through the R1 blade tip.

To provide a better understanding of the effects of changes in the upstream-to-downstream flow, we first present the velocity contour at the S2 vane entrance, as shown in Figure 11. It can be seen that the changes in the profile of the damaged blade had significant effects on the flow characteristics downstream. In the top-damage cases, the flow fields arriving into S2 were similar to those in the reference case. The flow extended from the hub—where the flow velocity was the highest—to the casing. The flow in the middle-damage cases also extended from the hub to the casing. However, the flow only developed to mid-span; the flow from 60%-span to the casing in the middle-damage cases was not significantly different from the corresponding flows in the top-damage cases and the reference case. This was due to the leakage from the various damaged rotor blade locations. In the top-damage cases, the leakage flow passing through the damage locations was not significant, resulting in flow velocity contours similar to those in the reference case. Conversely, the leakage flow in the middle-damage cases was noticeable, with a higher flow velocity around the mid-span. Moreover, the turbulence intensity of the flow increased due to changes in the blade profile resulting from the damage. The turbulence intensity also depended on the damage location—damage on the pressure side generated a higher turbulence intensity downstream than damage on the suction side did; in addition, damage at the middle created more turbulence than damage at the top did. This is because we had applied the HS for the inlet condition with the highest flow at the center, which had a more significant effect on the middle of the blade than that at the top.

From the velocity contours at the S2 vane entrance, shown in Figure 11, the flow arriving at the suction side of the S2 vane was more noticeable than that arriving at the pressure side. Figure 12 shows the velocity streamlines on the S2 vane suction side, which were used for analyzing the flow characteristics. The flow formation was less near the hub of the S2 vane in the middle-damage cases than in the reference case and top-damage cases, due to the higher tip leakage flow, which generated secondary flow and changed the flow structure in the passage and vane surface. Moreover, the flow trends near the shroud of the vane—denoted by the black rectangles in Figure 12—were strongly dependent on the damage locations. Compared with the reference case, there were fewer forming lines in the damage cases, due to the effects of damage on the secondary flow and tip leakage flow. Moreover, the difference in the non-uniform total pressure at the entrance of the S2 vane due to the effects of various damage locations resulted in changes in the flow structure on the S2 vane surface. Due to the effects of damage, compared with the reference case, the total pressure at the S2 vane entrance increased by 0.25% and 0.5% in the top-damage and middle-damage cases, respectively. The changes in the rotor blade profile

affected not only the passage flow but also the flow on the blade and vane surfaces due to their effects on the total pressure and consequently, on the leakage flow through the blade tip and passage.

**Figure 11.** Velocity contours at the entrance of the S2 vane.

**Figure 12.** Velocity streamlines at the suction side of the S2 vane surface.

The structure of the flow strongly affects its characteristics and the heat transfer properties of the vane surface. Figure 13 shows the temperature contours on the S2 vane surface under various blade conditions. Unlike for the R1 blade, the damage resulted in significant changes in both the pressure and suction sides for the S2 vane. This vane received more leakage flow when the blades were damaged; hence, more flow arrived at S2. The greater flow produced higher temperatures on both the pressure and suction sides of the S2 vane, leading to a significant increase in both the average and maximum temperatures, as shown in Figure 14. The increments in the average and maximum temperatures were approximately 9 and 7 K, respectively. These changes are more noticeable than those for the R1 blade surface. This increment in the temperature of the vane surface generated a higher thermal stress, which consequently reduced the fatigue life of the vane. It can be concluded that the damage on the rotor blade had more significant effects downstream than at the blade surface. Overall, the damage on the rotor blade considerably affected the flow characteristics both in the passage and on the surface of the blade and vane.

**Figure 13.** Temperature distribution on the S2 vane surface.

**Figure 14.** Average and maximum temperature of the S2 vane surface under various blade conditions.
